In this section, we fix a non-linearity \(\mathsf {H}:\mathbb {R}^d \rightarrow \mathbb {R}\)  and an initial condition \(\psi :\mathbb {R}_{\ge 0}^d\rightarrow \mathbb {R}\)  satisfying (REF )-(REF ), and we use the classical Perron method to establish the existence of solutions to the Hamilton-Jacobi equation (REF ). We closely follow the arguments in Chapter 5 of [1]}. It will be convenient to fix a positive constant
\(K>\sup \big \lbrace \vert \mathsf {H}(y)\vert \mid y\in \mathbb {R}^d \text{ with } {\vert \hspace{-0.42502pt}\vert \hspace{-0.42502pt}\vert y\vert \hspace{-0.42502pt}\vert \hspace{-0.42502pt}\vert }_{1,*}\le {\vert \hspace{-0.42502pt}\vert \hspace{-0.42502pt}\vert \psi \vert \hspace{-0.42502pt}\vert \hspace{-0.42502pt}\vert }_{\mathrm {Lip},1}\big \rbrace ,\) 
